## Elementary Linear Algebra 7th Edition

(a) The unit matrix $I_3$, that is $$I_3=\left[\begin{array}{ll}{1} & {0} &{0}\\ {0} & {1}&{0}\\{0}&{0}&{1}\end{array}\right].$$ (b) The matrices on the form $$\left[\begin{array}{ll}{1} & {0} &{a}\\ {0} & {1}&{b}\\{0}&{0}&{0}\end{array}\right], \quad a,b\in R.$$ For example, the matrix $$\left[\begin{array}{ll}{1} & {0} &{3}\\ {0} & {1}&{-2}\\{0}&{0}&{0}\end{array}\right].$$ (c) The matrices on the form $$\left[\begin{array}{ll}{1} & {a} &{b}\\ {0} & {0}&{0}\\{0}&{0}&{0}\end{array}\right],\quad a,b\in R$$ For example, the matrix $$\left[\begin{array}{ll}{1} & {-6} &{-5}\\ {0} & {0}&{0}\\{0}&{0}&{0}\end{array}\right].$$
The possible $3\times 3$ reduced row-echelon matrices can be one of the following forms: (a) The unit matrix $I_3$, that is $$I_3=\left[\begin{array}{ll}{1} & {0} &{0}\\ {0} & {1}&{0}\\{0}&{0}&{1}\end{array}\right].$$ (b) The matrices on the form $$\left[\begin{array}{ll}{1} & {0} &{a}\\ {0} & {1}&{b}\\{0}&{0}&{0}\end{array}\right], \quad a,b\in R.$$ For example, the matrix $$\left[\begin{array}{ll}{1} & {0} &{3}\\ {0} & {1}&{-2}\\{0}&{0}&{0}\end{array}\right].$$ (c) The matrices on the form $$\left[\begin{array}{ll}{1} & {a} &{b}\\ {0} & {0}&{0}\\{0}&{0}&{0}\end{array}\right],\quad a,b\in R$$ For example, the matrix $$\left[\begin{array}{ll}{1} & {-6} &{-5}\\ {0} & {0}&{0}\\{0}&{0}&{0}\end{array}\right].$$